# Abelian variety isogeny classes downloaded from the LMFDB on 24 April 2024.
# Search link: https://www.lmfdb.org/Variety/Abelian/Fq/1/2/
# Query "{'q': 2, 'g': 1}" returned 5 classes, sorted by dimension.
# Each entry in the following data list has the form:
# [Label, Dimension, Base field, L-polynomial, $p$-rank, Isogeny factors]
# For more details, see the definitions at the bottom of the file.
"1.2.ac" 1 2 [1, -2, 2] 0 [["1.2.ac", 1]]
"1.2.ab" 1 2 [1, -1, 2] 1 [["1.2.ab", 1]]
"1.2.a" 1 2 [1, 0, 2] 0 [["1.2.a", 1]]
"1.2.b" 1 2 [1, 1, 2] 1 [["1.2.b", 1]]
"1.2.c" 1 2 [1, 2, 2] 0 [["1.2.c", 1]]
#Dimension (g) --
# The **dimension** of an algebraic variety $V$ is the maximal length $d$ of a chain
# $$
# V_0 \subset V_1 \subset \cdots \subset V_d
# $$
# of distinct irreducible subvarieties of $V$.
#Base field (q) --
# The **base field**, of an algebraic variety is the field over which it is defined; it necessarily contains the coefficients of a set of defining equations for the variety, but it is not necessarily a minimal field of definition.
#L-polynomial (polynomial) --
# Let $A$ be an abelian variety of dimension $g$ defined over $\F_q$. Let $F_q$ be the inverse of the field automorphism $x \mapsto x^q$ in $\Gal(\overline{\F}_q/\F_q)$, which acts on $\ell$-adic \'etale cohomology. The **L-polynomial** of $A$ is
# $$L_A(t) = \det(1-t F_q|H^1(A_{\overline{\F}_q}, \Q_\ell)).$$
# This is a polynomial of degree $2g$ with integer coefficients that are independent of $\ell$. Its constant term is $1$.
# The L-polynomial $L_A(t)$ is the reverse of the characteristic polynomial $P_A(t)$, which is a Weil $q$-polynomial. Thus the complex roots of $L_A(t)$ have absolute value $q^{-1/2}$.
#$p$-rank (p_rank) --
# Let $A$ be a $g$-dimensional abelian variety over $\F_q$ where $q=p^r$.
# The **$p$-rank** of $A$ is the dimension of the geometric $p$-torsion as a $\F_p$-vector space: $$p\operatorname{-rank}(A) = \dim_{\F_p}( A(\overline{\F}_p)[p] ).$$ The $p$-rank is at most $g$, with equality if and only if $A$ is ordinary. The difference between $g$ and the $p$-rank is the **$p$-rank deficit** of $A$.
#Isogeny factors (decompositionraw) --
# Any abelian variety $A$ is isogenous to a product of simple abelian varieties $B_i$, called the **isogeny factors** of $A$:
# $$A \sim B_1 \times \cdots \times B_n.$$
# We say that $A$ decomposes up to isogeny into the product of the $B_i$.
# Note that two elements of this product might be isogenous; in other words, elements of the decomposition may appear with multiplicity.